Geodesic
Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions
Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1118-1130

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We consider the space $\mathcal C$ of conjugacy classes of the unitary group $\mathrm U(n+m)$ with respect to a smaller unitary group $\mathrm U(m)$. It is known that to every element of $\mathcal C$ we can canonically assign a rational matrix-valued function (the Livshits characteristic function) on the Riemann sphere. We find an explicit expression for the natural measure on $\mathcal C$ obtained as the push-forward of the Haar measure of $\mathrm U(n+m)$ in terms of characteristic functions.
Keywords: inner functions, characteristic functions, Haar measure, Cayley transform, random functions. 
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     author = {Yu. A. Neretin},
     title = {Radial parts of {Haar} measures and probability distributions on the space of rational matrix-valued functions},
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Yu. A. Neretin. Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions. Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1118-1130. http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a6/